(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given two sub-spaces of R^n - W_1 and W_2 where dimW_1 = dimW_2 =/= 0.

Prove that there exists an orthogonal transformation T:R^n -> R^n so that

T(W_1) = T(W_2)

2. Relevant equations

3. The attempt at a solution

If dimW_1 = dimW_2 = m then we can say that {v_1,...,v_m} is a basis of W_1 and {u_1,...,u_m} is a basis of W_2. We can make from these bases of R^n:

{v_1,...,v_m,a_1,...,a_{n-m}} is a basis of R^n and

{u_1,...,u_m,b_1,...,b_{n-m}} is also.

From these we can make orthonormal bases of R^n (via GS) so that:

{v'_1,...,v'_m,a'_1,...,a'_{n-m}} is an orthonormal basis of R^n and

{u'_1,...,u'_m,b'_1,...,b'_{n-m}} is also.

Now we make a transformation where:

T(v'_i) = u'_i

T(a'_i) = b'_i

Now, since the transformation of an orthonormal base gives another orthonormal base it's a ON transformation. And since {v'_1,...,v'_m} is a basis of W_1 and{u'_1,...,u'_m} is a basis of W_2 then T(W_1) = W_2

Is that a complete proof?

Thanks.

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# Orthogonal transformations

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